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A Compositional Framework for Passive Linear Networks | | John C. Baez
; Brendan Fong
; | | Date: |
22 Apr 2015 | | Abstract: | Passive linear networks are used in a wide variety of engineering
applications, but the best studied are electrical circuits made of resistors,
inductors and capacitors. We describe a category where a morphism is a circuit
of this sort with marked input and output terminals. In this category,
composition describes the process of attaching the outputs of one circuit to
the inputs of another. We construct a functor, dubbed the "black box functor",
that takes a circuit, forgets its internal structure, and remembers only its
external behavior. Two circuits have the same external behavior if and only if
they impose same relation between currents and potentials at their terminals.
The space of these currents and potentials naturally has the structure of a
symplectic vector space, and the relation imposed by a circuit is a Lagrangian
linear relation. Thus, the black box functor goes from our category of circuits
to the category of symplectic vector spaces and Lagrangian linear relations. We
prove that this functor is a symmetric monoidal dagger functor between dagger
compact categories. We assume the reader has some familiarity with category
theory, but none with circuit theory or symplectic linear algebra. | | Source: | arXiv, 1504.5625 | | Services: | Forum | Review | PDF | Favorites | |
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